Find all the critical numbers of the function f(x) = x+ln(|cos(x)|) that lie in the interval [pi, 3pi].
Please help, thanks in advance.
Hello, nubshat!
Exactly where is your difficulty?
$\displaystyle \text{Find all the critical numbers of the function: }\:f(x) \:=\: x\;+\;\ln(|\cos x|)\:\text{ in the interval }[\pi, 3\pi].$
Differentiate and equate to zero: .$\displaystyle f'(x) \:=\:1 + \frac{-\sin x}{\cos x} \:=\:0$
We have: .$\displaystyle 1 - \tan x \:=\:0 \quad\Rightarrow\quad \tan x \:=\:1$
Hence: .$\displaystyle x\:=\:\pm\tfrac{\pi}{4},\:\pm\tfrac{5\pi}{4},\:\pm \tfrac{9 \pi}{4},\:\pm\tfrac{13\pi}{4}\:\hdots$
Answers: .$\displaystyle x \;=\;\frac{5\pi}{4},\:\frac{9\pi}{4}$
Yes, I think both end points, too. Critical points are only used for finding minima and maxima, so assuming you're finding the minimum or maximum over the range $\displaystyle \pi$ to $\displaystyle 3\pi$, then you would need to include those points, too.
I suppose it could be that you're finding the minimum or maximum over a different range, and for some reason you're temporarily limiting yourself to finding critical points in the given range. That doesn't make much sense, though.
- Hollywood